Revista Matemática Iberoamericana

$m$-Berezin transform and compact operators

Kyesook Nam , Dechao Zheng , and Changyong Zhong

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Abstract

$m$-Berezin transforms are introduced for bounded operators on the Bergman space of the unit ball. The norm of the $m$-Berezin transform as a linear operator from the space of bounded operators to $L^{\infty}$ is found. We show that the $m$-Berezin transforms are commuting with each other and Lipschitz with respect to the pseudo-hyperbolic distance on the unit ball. Using the $m$-Berezin transforms we show that a radial operator in the Toeplitz algebra is compact iff its Berezin transform vanishes on the boundary of the unit ball.

Article information

Source
Rev. Mat. Iberoamericana, Volume 22, Number 3 (2006), 867-892.

Dates
First available in Project Euclid: 22 January 2007

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1169480034

Mathematical Reviews number (MathSciNet)
MR2320405

Zentralblatt MATH identifier
1125.47020

Subjects
Primary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15] 47B38: Operators on function spaces (general)

Keywords
$m$-Berezin transforms Toeplitz operators

Citation

Nam, Kyesook; Zheng, Dechao; Zhong, Changyong. $m$-Berezin transform and compact operators. Rev. Mat. Iberoamericana 22 (2006), no. 3, 867--892. https://projecteuclid.org/euclid.rmi/1169480034


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